An uncountable countable set
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From: Virgil on 14 Jul 2006 16:39 In article <1152883753.767753.301720(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > This definiion is > > > wrong. I show this by the fact that 0.111... is not in the set of unary > > > reprsetations of all natural numbers. aleph_0, the number of 1's in > > > 0.111... is *not* the cardinal number of |N. > > > > Proof, please. > > Define 0 *+ 0 = 0, 0 *+ 1 = 1 *+ 0 = 1 *+ 1 = 1. > Do the following addition *+ > > 0.1 > 0.11 > 0.111 > ... > > What is the result? 0.111... This number is not among the numbers to be > added. But it must be, if it is different from any list number and if > the result of the addition is 0.111... . Its absence does not matter as > its presence does not matter. Add by *+ > > 0,1 > 0.11 > 0.111 > ... > 0.111... > > The result is the same. Hence, if 0.111... exists, it is in he list and > is not in the list. What kind of objects have the property to be and > not to be? > > > > > > K cannot have infinitely many digits if all can be indexed by finite > > > numbers. Infinite is larger than any finite number. Therefore every > > > list number taken will not be capably of indexing K. > > > 0.111... - 0.111...1 = 0.000...0111... > > > > You mean covering here. And you are right, K is not in the list. I have > > never argued that it is. Still, what is the problem? > > What cannot be covered by naturals cannot be indexed by naturals. > > > > What do you mean with "cannot be indexed"? > > A position to be enumerated by a natural ordinal number (starting from > the first ,1) > > > > You use the word "indexed" in a strange way. You mean "covered". I never > > assume that 0.111... can be "covered" by some An. Nevertheless, *each > > digit position can be indexed by some An". > > As 0.111... consists of nothing else than 1's at each position, > covering of each position means covering of the whole number. Hence > your standpoint involves a contradiction. Only in the mind of "mueckenh". Since "mueckenh"'s 0.11... is a limit ordinal one only needs its successors to provide indexing to and beyond it. Since his notation is is so outre, I leave to him the problem of finding a suitable extension to that notation.
From: Dik T. Winter on 14 Jul 2006 19:34 In article <1152883753.767753.301720(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > This definiion is > > > wrong. I show this by the fact that 0.111... is not in the set of unary > > > reprsetations of all natural numbers. aleph_0, the number of 1's in > > > 0.111... is *not* the cardinal number of |N. > > > > Proof, please. > > Define 0 *+ 0 = 0, 0 *+ 1 = 1 *+ 0 = 1 *+ 1 = 1. > Do the following addition *+ > > 0.1 > 0.11 > 0.111 > ... How can I do that? You have not defined your operators on those objects. Yes, I know that you want to do it digit-wise on the digits. In that case there is still no definition for 0.1 *+ 0.11. So let us extend all numbers with zeros: 0.10... 0.110... 0.1110... so now we can go. But you have not defined how to do that "infinite sum". You use a lot of things for which you provide no defition. And, there is one natural number missing: 0.0... So let's see: {0.0...} has cardinality {0.10...} {0.0..., 0.10...} has cardinality {0.110...} .... > What is the result? 0.111... How do you get there? What rule are you using to calculate that "infinite sum"? But even if we allow that sum, we know that that is not the unary representation of a natural number. > This number is not among the numbers to be > added. But it must be, if it is different from any list number and if > the result of the addition is 0.111... . I see no reason for the "must be". Please first provide a proper definition of how that infinite operation should be performed. > Hence, if 0.111... exists, it is in he list and > is not in the list. It is not in the list and is not the unary representation of a natural number. So what? > > > K cannot have infinitely many digits if all can be indexed by finite > > > numbers. Infinite is larger than any finite number. Therefore every > > > list number taken will not be capably of indexing K. > > > 0.111... - 0.111...1 = 0.000...0111... > > > > You mean covering here. And you are right, K is not in the list. I have > > never argued that it is. Still, what is the problem? > > What cannot be covered by naturals cannot be indexed by naturals. Makes no sense to me. I did state: For all p there is an n such that An[p] = K[p]. You said that was wrong. I asked you why that was wrong, and you come up with some nonsense. I state: select n = p. Isn't it a fact that For all n, An[n] = K[n]? > > What do you mean with "cannot be indexed"? > > A position to be enumerated by a natural ordinal number (starting from > the first ,1) What do you mean with "A position to be enumerated"? You may indeed enumerate a single position, but when (as you state we have to start from 1) we get always 1 as result. So you must mean something else. Moreover, I asked yuo what "cannot be indexed" means and you answer with something positive. So this is a non-answer. > > You use the word "indexed" in a strange way. You mean "covered". I never > > assume that 0.111... can be "covered" by some An. Nevertheless, *each > > digit position can be indexed by some An". > > As 0.111... consists of nothing else than 1's at each position, > covering of each position means covering of the whole number. Hence > your standpoint involves a contradiction. Pray explain. With "indexed" I use the common meaning of "indexed", i.e. assigned a natural number to (i.e. a particular An) where each digit position obtains is assigned a different natural number. And I see *no* contradiction. Prove it. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 14 Jul 2006 20:03 In article <1152884145.033413.104250(a)75g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > These words are uninteresting. Transpositions or exchanges of two > > > elements are clearly defined. Infinitely many are admitted by Cantor. > > > > This is wrong. Not any sequence of transpositions is permitted. What > > *is* permitted is *transformations that can be written as a sequence of > > transpositions*. Because if you admit *any* sequence of transpositions > > you may destroy the order-type. > > Works, p. 214: Die Frage, durch welche Umformungen einer wohlgeordneten > Menge ihre Anzahl ge?ndert wird, durch welche nicht, l??t sich > einfach so beantworten, da? diejenigen und nur diejenigen Umformungen > die Anzahl unge?ndert lassen, welche sich zur?ckf?hren lassen auf > eine endliche oder unendliche Menge von Transpositionen, d. h. von > Vertauschungen je zweier Elemente. > > No word about sequence or permutation at this place. No, but still, not *any* set of transpositions is permitted. Only those sets of transpositions that are the result of rewriting a transformation ("Umformungen"). So it remains interesting what Cantor is meaning when he writes "Umformungen". And when he is meaning with that word arbitrary re-orderings, he is wrong (because *all* re-orderings can be rewritten as an infinite set of transpositions). > > > > The reordering of the naturals (0, 1, 2, ...) to (1, 2, 3, ..., 0) > > > > can be written as an infinite sequence of transpositions, but it > > > > changes the ordinal number. But is it a permutation? > > > > > > Unimportant for Cantor's statement. > > > > It *is* important. Because if that falls under Cantor's definition of > > "transformation" and "permutation", Cantor's statement is trivially false. > > Of course it is false, whether there is a sequence or not. My > transpositions form a sequence. One cannot do better. Yes, so what? Is yuor set of transpositions a rewriting of some "Umformung" in the sense Cantor implies? So there is no "of course" here. We need to know what Cantor means when he writes "Umformungen". > > Eh? You assume that Cantor's conclusion is true for your sequence of > > transpositions. I state that Cantor's conclusion is false for your > > sequence of statements. > > Cantor's conclusion is correct, with no doubt, if infinite sets like > the set of my transpositions do exist. No, when you assume that "Umformungen" means *any* re-ordering, it is trivially false if infinite sets do exist. But pray *prove* Cantor's conclusion based on (1) infinite sets do exist (2) an "Umformung" is any re-ordering You state there is no doubt, so it should be easy for you to provide a proof. > > > I know your interpretation is wrong. > > > > What part of my interpretation is wrong? > > The implication that Cantor's statement could be correct (if special > constraints were obeyed by permutations) with the implication that > infinity could exist. Can you provide a proof of that statement? Because you state just above: "Cantor's conclusion is correct, with no doubt, if infinite sets like the set of my transpositions do exist." Which contradicts what you state here. > > > But that does not at all affect my proof, because, as Virgil, > > > emphasized frequently, there is no transposition which changes the > > > well-order to normal order while maintaining the initial enumeration > > > (= well-order). Would infinitely many transpositions be possible, this > > > would necessarily happen. > > > > No. There would still not be a single transposition at which well-order > > changes to non well-order. As in lim{n -> oo} 1/n, there is not a > > single natural at which 1/n becomes 0. > > Therefore it never becomes 0 but always remains > 0 like 0.11... never > becomes 1/9. But apparently you do not use the standard definition of 0.111... . Is it a number or something else? If it is a number it will not become something, but it *is* something (numbers do not change while you look at them). Again, you have first to define *your* interpretation of 0.111... before we can even talk with sense about it. The standard definition is: 0.111... = sum{i = 1 .. oo} 10^(-i) = = lim{n -> oo} sum{i = 1 .. n} 10^(-i). What is your problem with that definition? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 14 Jul 2006 20:13 In article <1152882211.516443.276870(a)p79g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Randy Poe schrieb: > > If by "usual mathematics" you mean a mathematics in which the > > meaning of infinite decimals have not been defined, then we > > don't have 0.999... or 9.999... either. > > In usual mathematics 0.999... is assumed to exist. It is not assumed. There is a definition floating around giving a meaning to that notation. Without the definition that notation makes no sense. You are confusing notation with the basic concept. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 15 Jul 2006 11:49
Dik T. Winter schrieb: > > In usual mathematics 0.999... is assumed to exist. > > It is not assumed. There is a definition floating around giving a meaning > to that notation. > Without the definition that notation makes no sense. And with that definition it makes no sense either, as the definition is of he same kind as the squared circle. > You are confusing notation with the basic concept. The notation is the expression of the concept but it is impossible to separate both. Without an expression there is no concept. Both are assumed to exist, erroneously. Regards, WM |